Relative motion physics. KS. Relativity of motion. Relativity of path, trajectory and speed

If, in calm weather, a passenger who wakes up in the cabin of a sailing yacht looks out the window, he will not immediately understand whether the ship is sailing or adrift. Behind the thick glass is the monotonous surface of the sea, above is the blue sky with motionless clouds. However, in any case, the yacht will be in motion. And moreover, in several movements at once in relation to different reference systems. Even without taking into account the cosmic scale, this person, being at rest relative to the hull of the yacht, finds himself in a state of motion relative to the mass of water surrounding him. This can be seen in the wake. But even if the yacht is drifting with the sail lowered, it moves with the water flow that forms the sea current.

Thus, any body that is at rest relative to one body (reference system) is simultaneously in a state of motion relative to another body (another reference system).

Galileo's principle of relativity

Medieval scientists already thought about the relativity of motion, and in the Renaissance these ideas were further developed. “Why don’t we feel the Earth’s rotation?” – the thinkers wondered. Galileo Galilei gave a clear formulation based on physical laws to the principle of relativity. “For objects captured by uniform motion,” the scientist concluded, “this latter does not seem to exist and manifests its effect only on things that do not take part in it.” True, this statement is valid only within the framework of the laws of classical mechanics.

Relativity of path, trajectory and speed

The distance traveled, trajectory and speed of a body or point will also be relative depending on the chosen reference system. Take the example of the man walking through the carriages. His path over a certain period of time relative to the train will be equal to the distance traveled by his own feet. The path will consist of the distance traveled and the distance directly traveled by the person, regardless of which direction he walked. The same with speed. But here the speed of a person’s movement relative to the ground will be higher than the speed of movement - if the person is walking in the direction of the train, and lower - if he is walking in the opposite direction to the movement.

It is convenient to trace the relativity of the trajectory of a point using the example of a nut attached to the rim of a bicycle wheel and holding a spoke. It will be motionless relative to the rim. Relative to the body of the bicycle, this will be the trajectory of a circle. And relative to the ground, the trajectory of this point will be a continuous chain of semicircles.

I suggest a game: choose an object in the room and describe its location. Do this in such a way that the guesser cannot make a mistake. Did it work out? What will come of the description if other bodies are not used? The following expressions will remain: “to the left of...”, “above...” and the like. Body position can only be set relative to some other body.

Location of the treasure: “Stand at the eastern corner of the outermost house, face north and, having walked 120 steps, turn to face east and walk 200 steps. In this place, dig a hole 10 cubits in size and you will find 100 gold bars.” It is impossible to find the treasure, otherwise it would have been dug up long ago. Why? The body in relation to which the description is being made is not defined; it is unknown in which village that very house is located. It is necessary to accurately determine the body that will serve as the basis for our future description. In physics such a body is called reference body. It can be selected arbitrarily. For example, try to choose two different reference bodies and describe the location of a computer in a room relative to them. There will be two descriptions that are different from each other.

Coordinate system

Let's look at the picture. Where is the tree relative to cyclist I, cyclist II, and us looking at the monitor?

Relative to the reference body - cyclist I - the tree is on the right, relative to the reference body - cyclist II - the tree is on the left, relative to us it is in front. One and the same body - a tree, constantly located in the same place, at the same time “to the left”, and “to the right” and “in front”. The problem is not only that different reference bodies are chosen. Let's consider its location relative to cyclist I.

In this picture there is a tree on right from cyclist I

In this picture there is a tree left from cyclist I

The tree and the cyclist did not change their location in space, but the tree can be “on the left” and “on the right” at the same time. In order to get rid of the ambiguity in the description of the direction itself, we will choose a certain direction as positive, the opposite of the chosen one will be negative. The selected direction is indicated by an axis with an arrow, the arrow indicating the positive direction. In our example, we will select and designate two directions. From left to right (the axis along which the cyclist moves), and from us inside the monitor to the tree - this is the second positive direction. If the first direction we have chosen is designated as X, the second - as Y, we obtain a two-dimensional coordinate system.

Relative to us, the cyclist is moving in a negative direction along the X axis, the tree is in a positive direction along the Y axis

Relative to us, the cyclist is moving in the positive direction along the X axis, the tree is in the positive direction along the Y axis

Now determine which object in the room is 2 meters in the positive X direction (to your right), and 3 meters in the negative Y direction (behind you). (2;-3) - coordinates this body. The first number “2” usually indicates the location along the X axis, the second number “-3” indicates the location along the Y axis. It is negative because the Y axis is not on the side of the tree, but on the opposite side. After the body of reference and direction is selected, the location of any object will be described unambiguously. If you turn your back to the monitor, there will be another object to the right and behind you, but its coordinates will be different (-2;3). Thus, the coordinates accurately and unambiguously determine the location of the object.

The space in which we live is a space of three dimensions, as they say, three-dimensional space. In addition to the fact that the body can be “to the right” (“left”), “in front” (“behind”), it can also be “above” or “below” you. This is the third direction - it is customary to designate it as the Z axis

Is it possible to choose different axis directions? Can. But you cannot change their directions while solving, for example, one problem. Can I choose other axis names? It is possible, but you risk that others will not understand you; it is better not to do this. Is it possible to swap the X axis with the Y axis? You can, but don't get confused about the coordinates: (x;y).

When a body moves in a straight line, one coordinate axis is sufficient to determine its position.

To describe movement on a plane, a rectangular coordinate system is used, consisting of two mutually perpendicular axes (Cartesian coordinate system).

Using a three-dimensional coordinate system, you can determine the position of a body in space.

Reference system

Each body at any moment of time occupies a certain position in space relative to other bodies. We already know how to determine its position. If the position of a body does not change over time, then it is at rest. If the position of the body changes over time, this means that the body is moving. Everything in the world happens somewhere and sometime: in space (where?) and in time (when?). If we add a method of measuring time - a clock - to the reference body, the coordinate system that determines the position of the body, we get reference system. With the help of which you can evaluate whether a body is moving or at rest.

Relativity of motion

The astronaut went into outer space. Is it in a state of rest or movement? If we consider it relative to the cosmonaut's friend who is nearby, he will be at rest. And if relative to an observer on Earth, the astronaut is moving at enormous speed. Same with traveling on a train. Regarding the people on the train, you sit motionless and read a book. But relative to the people who stayed at home, you are moving at the speed of a train.

Examples of choosing a reference body, relative to which in figure a) the train is moving (relative to the trees), in figure b) the train is at rest relative to the boy.

Sitting in the carriage, we await departure. In the window we watch the train on a parallel track. When it starts to move, it is difficult to determine who is moving - our carriage or the train outside the window. In order to decide, it is necessary to evaluate whether we are moving relative to other stationary objects outside the window. We evaluate the state of our carriage relative to various reference systems.

Changing displacement and speed in different reference systems

Displacement and speed change when moving from one frame of reference to another.

The speed of a person relative to the ground (a fixed frame of reference) is different in the first and second cases.

Rule for adding speeds: The speed of a body relative to a fixed frame of reference is the vector sum of the speed of the body relative to a moving frame of reference and the speed of the moving frame of reference relative to a stationary one.

Similar to the displacement vector. Rule for adding movements: The displacement of a body relative to a fixed reference system is the vector sum of the displacement of the body relative to a moving reference system and the displacement of a moving reference system relative to a stationary one.

Let a person walk along the carriage in the direction (or against) the movement of the train. Man is a body. The earth is a fixed frame of reference. The carriage is a moving frame of reference.

Changing trajectory in different reference systems

The trajectory of a body's movement is relative. For example, consider the propeller of a helicopter descending to Earth. A point on the propeller describes a circle in the reference frame associated with the helicopter. The trajectory of this point in the reference frame associated with the Earth is a helical line.

Forward movement

The movement of a body is a change in its position in space relative to other bodies over time. Each body has certain dimensions, sometimes different points of the body are in different places in space. How to determine the position of all points of the body?

BUT! Sometimes it is not necessary to indicate the position of every point on the body. Let's consider similar cases. For example, this does not need to be done when all points of the body move the same way.

All the currents of the suitcase and car move the same way.

The movement of a body in which all its points move equally is called progressive

Material point

There is no need to describe the movement of each point of the body even when its dimensions are very small compared to the distance it travels. For example, a ship crossing the ocean. When describing the motion of planets and celestial bodies relative to each other, astronomers do not take into account their sizes and their own motion. Despite the fact that, for example, the Earth is huge, relative to the distance to the Sun it is negligible.

There is no need to consider the movement of each point of the body when they do not affect the movement of the entire body. Such a body can be represented by a point. It’s as if we concentrate all the substance of the body into a point. We get a model of the body, without dimensions, but it has mass. That's what it is material point.

The same body with some of its movements can be considered a material point, but with others it cannot. For example, when a boy walks from home to school and at the same time covers a distance of 1 km, then in this movement he can be considered a material point. But when the same boy performs exercises, he can no longer be considered a point.

Consider moving athletes

In this case, the athlete can be modeled by a material point

In the case of an athlete jumping into water (picture on the right), it is impossible to model it to a point, since the movement of the entire body depends on any position of the arms and legs

The main thing to remember

1) The position of the body in space is determined relative to the reference body;
2) It is necessary to specify the axes (their directions), i.e. a coordinate system that defines the coordinates of the body;
3) The movement of the body is determined relative to the reference system;
4) In different reference systems, the speed of a body can be different;
5) What is a material point

A more complex situation of adding speeds. Let a man cross a river in a boat. The boat is the body under study. The fixed frame of reference is the earth. The moving frame of reference is the river.

The boat's speed relative to the ground is a vector sum

What is the displacement of any point located on the edge of a disk of radius R when it is rotated relative to the stand by 600? at 1800? Solve in the frames of reference associated with the stand and the disk.

In the reference frame associated with the stand, the displacements are R and 2R. In the reference frame associated with the disk, the displacement is zero all the time.

Why do raindrops in calm weather leave inclined straight stripes on the windows of a uniformly moving train?

In the reference frame associated with the Earth, the trajectory of the drop is a vertical line. In the frame of reference associated with the train, the movement of a drop on the glass is the result of the addition of two rectilinear and uniform movements: the train and the uniform fall of the drop in the air. Therefore, the trail of a drop on glass is inclined.

How can you determine your running speed if you train on a treadmill with a broken automatic speed detection? After all, you can’t move a single meter relative to the walls of the hall.

Is it possible to be stationary and still move faster than a Formula 1 car? It turns out that it is possible. Any movement depends on the choice of reference system, that is, any movement is relative. The topic of today's lesson: “Relativity of motion. The law of addition of displacements and velocities." We will learn how to choose a reference system in a given case, and how to find the displacement and velocity of a body.

Mechanical motion is the change in the position of a body in space relative to other bodies over time. The key phrase in this definition is “relative to other bodies.” Each of us is motionless relative to any surface, but relative to the Sun we, together with the entire Earth, undergo orbital motion at a speed of 30 km/s, that is, the motion depends on the reference system.

A reference system is a set of coordinate systems and clocks associated with the body relative to which motion is being studied. For example, when describing the movements of passengers inside a car, the reference system can be associated with a roadside cafe, or with the inside of a car, or with a moving oncoming car if we are estimating the overtaking time (Fig. 1).

Rice. 1. Selection of reference system

What physical quantities and concepts depend on the choice of reference system?

1. Body position or coordinates

Let's consider an arbitrary point. In different systems it has different coordinates (Fig. 2).

Rice. 2. Coordinates of a point in different coordinate systems

2. Trajectory

Consider the trajectory of a point on an airplane propeller in two reference frames: the reference frame associated with the pilot, and the reference frame associated with the observer on Earth. For the pilot, this point will perform a circular rotation (Fig. 3).

Rice. 3. Circular rotation

While for an observer on Earth the trajectory of this point will be a helical line (Fig. 4). Obviously, the trajectory depends on the choice of reference system.

Rice. 4. Helical path

Relativity of trajectory. Trajectories of body motion in various reference systems

Let's consider how the trajectory of movement changes depending on the choice of reference system using the example of a problem.

Task

What will be the trajectory of the point at the end of the propeller in different reference points?

1. In the CO associated with the pilot of the aircraft.

2. In the CO associated with the observer on Earth.

Solution:

1. Neither the pilot nor the propeller moves relative to the airplane. For the pilot, the trajectory of the point will appear to be a circle (Fig. 5).

Rice. 5. Trajectory of the point relative to the pilot

2. For an observer on Earth, a point moves in two ways: rotating and moving forward. The trajectory will be helical (Fig. 6).

Rice. 6. Trajectory of a point relative to an observer on Earth

Answer : 1) circle; 2) helix.

Using this problem as an example, we were convinced that trajectory is a relative concept.

As an independent test, we suggest you solve the following problem:

What will be the trajectory of a point at the end of the wheel relative to the center of the wheel, if this wheel moves forward, and relative to points on the ground (a stationary observer)?

3. Movement and path

Let's consider a situation when a raft is floating and at some point a swimmer jumps off it and tries to cross to the opposite shore. The movement of the swimmer relative to the fisherman sitting on the shore and relative to the raft will be different (Fig. 7).

Movement relative to the ground is called absolute, and relative to a moving body - relative. The movement of a moving body (raft) relative to a stationary body (fisherman) is called portable.

Rice. 7. Swimmer's movement

From the example it follows that displacement and path are relative quantities.

4. Speed

Using the previous example, you can easily show that speed is also a relative quantity. After all, speed is the ratio of movement to time. Our time is the same, but our travel is different. Therefore, the speed will be different.

The dependence of the characteristics of motion on the choice of reference system is called relativity of motion.

In the history of mankind, there have been dramatic cases associated precisely with the choice of a reference system. The execution of Giordano Bruno, the abdication of Galileo Galilei - all these are consequences of the struggle between supporters of the geocentric frame of reference and the heliocentric frame of reference. It was very difficult for humanity to get used to the idea that the Earth is not the center of the universe at all, but a completely ordinary planet. And movement can be considered not only relative to the Earth, this movement will be absolute and relative to the Sun, stars or any other bodies. Describing the motion of celestial bodies in a reference frame associated with the Sun is much more convenient and simpler; this was convincingly shown first by Kepler, and then by Newton, who, based on a consideration of the motion of the Moon around the Earth, derived his famous law of universal gravitation.

If we say that the trajectory, path, displacement and speed are relative, that is, they depend on the choice of the reference system, then we do not say this about time. Within the framework of classical, or Newtonian, mechanics, time is an absolute value, that is, it flows equally in all reference systems.

Let's consider how to find displacement and velocity in one reference system if they are known to us in another reference system.

Let's consider the previous situation, when a raft is floating and at some point a swimmer jumps off it and tries to cross to the opposite shore.

How is the movement of a swimmer relative to a stationary SO (associated with the fisherman) connected with the movement of a relatively mobile SO (associated with the raft) (Fig. 8)?

Rice. 8. Illustration for the problem

We called movement in a stationary frame of reference . From the vector triangle it follows that . Now let's move on to finding the relationship between speeds. Let us remember that within the framework of Newtonian mechanics, time is an absolute value (time flows the same in all reference systems). This means that each term from the previous equality can be divided by time. We get:

This is the speed at which a swimmer moves for a fisherman;

This is the swimmer's own speed;

This is the speed of the raft (the speed of the river).

Problem on the law of addition of velocities

Let's consider the law of adding velocities using an example problem.

Task

Two cars are moving towards each other: the first car at speed , the second at speed . At what speed are the cars approaching each other (Fig. 9)?

Rice. 9. Illustration for the problem

Solution

Let us apply the law of addition of velocities. To do this, let's move from the usual CO associated with the Earth to CO associated with the first car. Thus, the first car becomes stationary, and the second one moves towards it with speed (relative speed). At what speed, if the first car is stationary, does the Earth rotate around the first car? It rotates at a speed and the speed is directed in the direction of the speed of the second car (transfer speed). Two vectors that are directed along the same straight line are summed. .

Answer: .

Limits of applicability of the law of addition of velocities. The law of addition of velocities in the theory of relativity

For a long time it was believed that the classical law of addition of velocities is always valid and applies to all reference systems. However, about years ago it turned out that in some situations this law does not work. Let's consider this case using an example problem.

Imagine that you are on a space rocket moving at a speed of . And the captain of the space rocket turns on the flashlight in the direction of the rocket's movement (Fig. 10). The speed of light propagation in vacuum is . What will be the speed of light for a stationary observer on Earth? Will it be equal to the sum of the speeds of light and the rocket?

Rice. 10. Illustration for the problem

The fact is that here physics is faced with two contradictory concepts. On the one hand, according to Maxwell's electrodynamics, the maximum speed is the speed of light, and it is equal to . On the other hand, according to Newtonian mechanics, time is an absolute value. The problem was solved when Einstein proposed the special theory of relativity, or rather its postulates. He was the first to suggest that time is not absolute. That is, somewhere it flows faster, and somewhere slower. Of course, in our world of low speeds we do not notice this effect. In order to feel this difference, we need to move at speeds close to the speed of light. Based on Einstein's conclusions, the law of addition of velocities in the special theory of relativity was obtained. It looks like this:

This is the speed relative to a stationary CO;

This is the speed of relatively mobile CO;

This is the speed of the moving CO relative to the stationary CO.

If we substitute the values ​​from our problem, we find that the speed of light for a stationary observer on Earth will be .

The controversy has been resolved. You can also make sure that if the velocities are very small compared to the speed of light, then the formula for the theory of relativity turns into the classical formula for adding velocities.

In most cases we will use the classical law.

Today we found out that movement depends on the reference system, that speed, path, movement and trajectory are relative concepts. And time, within the framework of classical mechanics, is an absolute concept. We learned to apply the acquired knowledge by analyzing some typical examples.

Bibliography

1. Tikhomirova S.A., Yavorsky B.M. Physics (basic level) - M.: Mnemosyne, 2012.
2. Gendenshtein L.E., Dick Yu.I. Physics 10th grade. - M.: Mnemosyne, 2014.
3. Kikoin I.K., Kikoin A.K. Physics - 9, Moscow, Education, 1990.

1. Internet portal Class-fizika.narod.ru ().
2. Internet portal Nado5.ru ().
3. Internet portal Fizika.ayp.ru ().

Homework

1. Define the relativity of motion.
2. What physical quantities depend on the choice of reference system?

Imagine an electric train. She travels quietly along the rails, transporting passengers to their dachas. And suddenly, sitting in the last carriage, the hooligan and parasite Sidorov notices that at the Sady station controllers are entering the carriage. Naturally, Sidorov did not buy a ticket, and he wants to pay the fine even less.

Relativity of free rider movement on a train

And so, to avoid being caught, he quickly moves into another carriage. The controllers, having checked the tickets of all passengers, move in the same direction. Sidorov again moves to the next carriage and so on.

And so, when he reaches the first carriage and there is nowhere to go further, it turns out that the train has just reached the Ogorody station he needs, and happy Sidorov gets out, rejoicing that he rode like a hare and did not get caught.

What can we learn from this action-packed story? We can, without a doubt, rejoice for Sidorov, and we can, in addition, discover another interesting fact.

While the train traveled five kilometers from the Sady station to the Ogorody station in five minutes, the Sidorov hare covered the same distance plus a distance equal to the length of the train in which it was traveling, that is, about five thousand two hundred meters in the same five minutes.

It turns out that Sidorov was moving faster than the train. However, the controllers following on his heels developed the same speed. Considering that the train speed was about 60 km/h, it was time to give them all several Olympic medals.

However, of course, no one will engage in such stupidity, because everyone understands that Sidorov’s incredible speed was developed by him only relative to stationary stations, rails and vegetable gardens, and this speed was determined by the movement of the train, and not at all by Sidorov’s incredible abilities.

In relation to the train, Sidorov was not moving fast at all and did not even reach the Olympic medal, but even the ribbon from it. This is where we come across such a concept as the relativity of motion.

The concept of relativity of motion: examples

The relativity of motion has no definition, since it is not a physical quantity. The relativity of mechanical movement is manifested in the fact that some characteristics of movement, such as speed, path, trajectory, and so on, are relative, that is, they depend on the observer. In different reference systems these characteristics will be different.

In addition to the example given with citizen Sidorov on the train, you can take almost any movement of any body and show how relative it is. When going to work, you move forward relative to your house and at the same time move backward relative to the bus you missed.

You stand still relative to the player in your pocket and rush at great speed relative to a star called the Sun. Every step you take will be a gigantic distance for an asphalt molecule and insignificant for the planet Earth. Any movement, like all its characteristics, always makes sense only in relation to something else.

There is also a provision in the school curriculum that any movement of one body can only be recorded relative to another body. This position is called the term “relativity of motion.” From the pictures in the textbooks, it was clear that for someone standing on the river bank, a boat floating past consists of its speed and the speed of the river current. After such a detailed consideration, it becomes clear that the relativity of motion surrounds us in all aspects of our lives. The speed of an object is a relative quantity, but its derivative, acceleration, also becomes. The importance of this conclusion lies in the fact that it is acceleration that is included in the formula of Newton’s second law (the fundamental law of mechanics). According to this law, any force acting on a body gives it an acceleration proportional to it. The relativity of motion forces us to ask an additional question: relative to what body is acceleration given?

This law does not contain any explanations on this matter, but through simple logical deductions one can come to the conclusion that since force is a measure of the influence of one body (1) on another (2), then this same force imparts acceleration to the body (2) relative to the body (1), and not just some abstract acceleration.

The relativity of motion is the dependence of a certain body, a certain path, speed and movement on the selected reference systems. In terms of kinematics, any reference systems used are equal, but at the same time all the kinematic characteristics of this movement (trajectory, speed, displacement) are different in them. All quantities that depend on the chosen reference system with which they will be measured are called relative.

The relativity of motion, which is quite difficult to define without a detailed consideration of other concepts, requires precise mathematical calculations. We can talk about whether a body is moving or not when it is absolutely clear relative to what (the reference body) its position is changing. The reference system is a set of elements such as the reference body, as well as the coordinate systems and time reference systems associated with it. In relation to these elements, the movement of any bodies is considered or Mathematically, the movement of an object (point) in relation to the chosen reference system is described by equations that establish how the coordinates that determine the position of the object in this system change in time. Such equations that determine the relativity of motion are called equations of motion.

In modern mechanics, any movement of an object is relative, so it should be considered only in relation to another object (a reference body) or an entire system of bodies. For example, you cannot simply point out that the Moon moves at all. The correct statement would be that the Moon moves in relation to the Sun, Earth, and stars.

Often in mechanics, the reference system is linked not to the body, but to a whole continuum of basic bodies (real or fictitious) that define the coordinate system.

Movies often show motion relative to various bodies. So, for example, in some frames they show a train moving against the backdrop of some landscape (this is movement relative to the surface of the Earth), and in the next - a compartment of a carriage with trees flashing through the windows (movement relative to one carriage). Any movement or rest of a body, which is a special case of movement, is relative. Therefore, when answering a simple question whether a body is moving or at rest, and how it moves, it is necessary to clarify in relation to which objects its movement is being considered. The choice of reference systems, as a rule, is made depending on the stated conditions of the problem.